Rings
Table of Contents

Rings

Recall from the Groups page that a group is a set $G$ with a binary operation $*$ denoted $(G, *)$ such that:

  • 1. $G$ is closed under $*$, that is, for all $a, b \in G$ we have that $(a * b) \in G$.
  • 2. The operation $*$ is associative, that is, for all $a, b, c \in G$ we have that $a * (b * c) = (a * b) * c$.
  • 3. There exists an element $e \in G$ such that for all $a \in G$ we have that $a * e = a$ and $e * a = a$ to which we call $e$ the unique identity of $*$ in $G$.
  • 4. For each $a \in S$ there exists an $a^{-1} \in G$ such that $a * a^{-1} = e$ and $a^{-1} * a = e$ to which we call $a^{-1}$ the inverse of $a$.

We will now briefly describe another type of algebraic structure known as a ring.

Definition: If $+$ and $*$ are binary operations on the set $R$, then $R$ is called a Ring under $+$ and $*$ denoted $(R, +, *)$ if $R$ under $+$ and $*$ satisfies the following properties:
1. For all $a, b \in R$ we have that $(a + b \in R)$ (Closure under $+$).
2. For all $a, b, c \in R$, $a + (b + c) = (a + b) + c$ (Associativity of elements in $R$ under $+$).
3. There exists an $0 \in R$ such that for all $a \in R$ we have that $a + 0 = a$ and $0 + a = a$ (The existence of an identity element $0$ of $R$ under $+$).
4. For all $a \in R$ there exists a $-a \in R$ such that $a + (-a) = 0$ and $(-a) + a = 0$ (The existence of inverses for each element in $R$ under $+$).
5. For all $a, b \in R$ we have that $a + b = b + a$ (Commutativity of elements in $R$ under $+$).
6. For all $a, b \in R$ we have that $a * b = b * a$ (Closure under $*$).
7. For all $a, b, c \in R$, $a * (b * c) = (a * b) * c$ (Associativity of elements in $R$ under $*$).
8. There exists a $1 \in R$ such that for all $a \in R$ we have hat $a * 1 = a$ and $1 * a = a$ (The existence of an identity element $1$ of $R$ under $*$).
9. For all $a, b, c \in R$ we have that $a * (b + c) = (a * b) + (b * c)$ and $(a + b) * c = (a * c) + (b * c)$ (Distributivity of $*$ over $+$).

The operation $+$ is commonly referred to as addition while the operation $*$ is commonly referred to as multiplication. In the definition above, instead of using $e$, we use $0$ to denote the identity element of the operation $+$ commonly referred to as the Additive Identity, and we use $1$ to denote the identity element of the operation $*$ commonly referred to as the Multiplicative Identity.

Some people define a ring with the axioms 1-7 and 9 while omitting 8 and define a "ring with identity" or "ring with unit" to be the structure that satisfies the axioms 1-9. For our purposes, we will define a ring to always have a multiplicative identity.

Note that the first four conditions for $(R, +, *)$ to be a ring are the conditions for $(R, +)$ to be a group. Therefore, every ring is a group with the difference being that a group contains a second operation $*$ that is closed, associative, contains an identity element, and distributes over $+$.

One example of a ring in the set of rational numbers $\mathbb{Q}$ with the operations $+$ of standard addition and $*$ of standard multiplication. Let's verify this.

It's easy to verify that $\mathbb{Q}$ is closed under $+$ and that $+$ is associative. Furthermore, the additive identity is $0 \in \mathbb{Q}$. For each $x \in \mathbb{Q}$ we have that $x = \frac{a}{b}$ for $a, b \in \mathbb{Z}$ and $b \neq 0$ and the additive inverse of $x$ is $-x = - \frac{a}{b} \in \mathbb{Q}$ since $\displaystyle{x + (-x) = \frac{a}{b} + \left ( -\frac{a}{b} \right ) = 0}$. Furthermore, it is obvious that the addition of rational numbers is commutative.

If $x, y \in \mathbb{Q}$ where $\displaystyle{x = \frac{a}{b}}$ and $\displaystyle{y = \frac{c}{d}}$ with $a, b, c, d \in \mathbb{Z}$ and $b, d \neq 0$ then the product of $a$ and $b$ is closed under $*$ since $ac, bd \in \mathbb{Z}$ and $bd \neq 0$:

(1)
\begin{align} \quad x * y = \frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd} \in \mathbb{Q} \end{align}

It can be easily verified that $*$ is an associated operation for addition of rational numbers.

The multiplicative identity is the rational number $\displaystyle{1 = \frac{1}{1}}$

Lastly, it's not hard to show that the distributivity property holds. Let $x = \frac{a}{b}, y = \frac{c}{d}, z=\frac{e}{f} \in \mathbb{Q}$. For left distributivity we have:

(2)
\begin{align} \quad x * (y + z) = \frac{a}{b} * \left ( \frac{c}{d} + \frac{e}{f} \right ) = \frac{ac}{bd} + \frac{ae}{bf} = x *y + x * z \end{align}

For right distributivity we have:

(3)
\begin{align} \quad (x + y) * z = \left ( \frac{a}{b} + \frac{c}{d} \right ) * \frac{e}{f} = \frac{ae}{bf} + \frac{ce}{df} = x*z + y*z \end{align}

Therefore $(\mathbb{Q}, +, *)$ is a ring.

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